Download Chapter 7 notes physics 2

Chapter 7 – Impulse and Momentum
7.1 – The Impulse – Momentum Theorem
Definition of Impulse
The impulse J of a force is the product of the average force ̅ and the time interval t during
which the force acts:
̅
Impulse is a vector quantity and has the same direction as the average force.
SI Unit of Impulse: newton∙second (N∙s)
A large impulse produces a large response.
Both mass and velocity play a role in how an object responds to a given impulse.
Definition of Linear Momentum
The linear momentum p of an object is the product of the object’s mass m and velocity v:
Linear momentum is a vector quantity that points in the same direction as the velocity.
SI Unit of Linear Momentum: kilogram∙meter/second (kg∙m/s)
Newton’s second law of motion can be used to reveal a relationship between impulse and
momentum.
The average acceleration of an object can be given by
̅
Using Newton’s second law ∑
̅, we can substitute in for ̅.
∑̅
(
)
Impulse –Momentum Theorem
When a net force acts on an object, the impulse of this force is equal to the change in
momentum of the object:
(∑ ̅ )
Impulse
Final
Initial
Momentum Momentum
Impulse = Change in momentum
7.2 – The Principle of Conservation of Linear Momentum
Comparing the impulse-momentum theorem to the work-energy theorem, we see that the
impulse-momentum theorem states that the impulse produced by a net force is equal to the
change in the object’s momentum, while the work –energy theorem states that the work done
by a net force is equal to the change in the objects’ kinetic energy.
Two types of forces that occur during a typical collision.
1. Internal forces – Forces that the objects within the system exert on each other.
2. External forces – Forces exerted on the objects by agents external to the system.
Typically during a collision we see that
(∑
∑
)
For an isolated system the total net force on the system is zero.
As a result
Principle of Conservation of Linear Momentum
The total linear momentum of an isolated system remains constant (is conserved). An isolated
system is one for which the vector sum of the average external forces acting on the system is
zero.
This principle applies to a system containing any number of objects regardless of the internal
forces, provided the system is isolated.
It is important to know that the total linear momentum may be conserved even when the
kinetic energies of the individual parts of a system change.
7.3 – Collisions in One Dimension
When two macroscopic objects collide, such as two cars, the total kinetic energy after the
collision is generally less than that before the collision.
Kinetic energy is lost mainly in two ways:
1. It can be converted into heat because of friction
2. It is spent in creating permanent distortion or damage
Collisions are often classified according to whether the total kinetic energy changes during the
collision:
1. Elastic collision – One in which the total kinetic energy of the system after collision is
equal to the total kinetic energy before the collision.
2. Inelastic collision – One in which the total kinetic energy of the system is not the same
before and after the collision; if the objects stick together after colliding , the collision is
said to be completely inelastic.
When working on collision problems in order to find specific answers you may have to use both
concepts (conservation of momentum and conservation of energy) in order to solve for your
variables.
For certain situations this can be difficult and time consuming. To simplify your efforts use the
given equations for the SPECIFIC situation.
Elastic collision:
has an initial velocity ( )
is initially at rest (
)
(
)
(
)
**In order to use this specific situation you must remember
starts from rest.
7.4 – Collisions in Two Dimensions
Momentum is a vector quantity, however, and in two dimensions the x and y components of
the total momentum are conserved separately.
x Component
Pfx
P0x
Pfy
P0y
y Component
If a system contains more than two objects, a mass-times-velocity term must be included for
each additional object on either side of the Equations.
7.5 – Center of Mass
The center of mass is a point that represents the average location for the total mass of a
system.
Center of mass
If a system contains more than two particles, the center-of-mass point can be determined by
including an additional mass and position for each new particle.
For a macroscopic object, which contains many, many particles, the center-of-mass point is
located at the geometric center of the object, provided that the mass is distribute
symmetrically about the center.
For a non-symmetric system, the center-of-mass point is not located at the geometric center.
Velocity of center of mass
If the total linear momentum of a system of particles remains constant during an interaction
such as a collision, the velocity of the center of mass also remains constant.